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Let's tackle the questions from the image you uploaded. 

### For problems 33 to 42:
They involve finding linear equations based on given conditions, points, or graphs.

#### Starting with question **33**:
You are asked to find a linear equation where:
- \( f(-5) = -4 \)
- \( f(5) = 2 \)

Using the **two-point form** of a line, the formula for a line through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
\[
y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)
\]

The two points based on the function values are:
- \( (-5, -4) \)
- \( (5, 2) \)

The slope (\( m \)) is calculated as:
\[
m = \frac{2 - (-4)}{5 - (-5)} = \frac{2 + 4}{5 + 5} = \frac{6}{10} = 0.6
\]

Using the point-slope form:
\[
y - (-4) = 0.6(x - (-5))
\]
Simplify:
\[
y + 4 = 0.6(x + 5)
\]
\[
y + 4 = 0.6x + 3
\]
\[
y = 0.6x - 1
\]

Thus, the equation for **33** is:
\[
y = 0.6x - 1
\]

#### For **41** (Graph):
This involves finding the equation of the line shown on the graph.

- The line passes through points \( (-4, -3) \) and \( (4, 3) \).

Using the same method (slope formula):
\[
m = \frac{3 - (-3)}{4 - (-4)} = \frac{3 + 3}{4 + 4} = \frac{6}{8} = 0.75
\]

Using the point-slope form with the point \( (-4, -3) \):
\[
y - (-3) = 0.75(x - (-4))
\]
Simplify:
\[
y + 3 = 0.75(x + 4)
\]
\[
y + 3 = 0.75x + 3
\]
\[
y = 0.75x
\]

Thus, the equation for **41** is:
\[
y = 0.75x
\]

Would you like me to continue solving the rest, or clarify any specific questions? Here are some relative questions:

1. How is the slope calculated between two points?
2. What is the point-slope form of a linear equation?
3. How can you verify a point lies on a given line?
4. What does it mean when the slope of a line is positive or negative?
5. Can you describe the steps to graph a linear equation given its slope and a point?

**Tip**: Always simplify the slope before plugging it into the point-slope form to avoid calculation errors.

Find an equation for the function graphed in problem 41 and problem 42.

Discover how to derive the equation of a line from its graph with step-by-step examples. This guide covers key algebraic concepts like the slope formula and point-slope form of a line, ideal for high school students learning coordinate geometry.

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